Track Groups (I)

A Track Group (P, Q)(X, xQ; x0) is, for a large class of spaces, the mth homotopy group of the function space F of maps of P into X which carry the closed (possibly empty) subset Q to x0, a point of X; the base point is, in general, the trivial map x0, such that xo(P) = xQ. These homotopy groups of function spaces have been studied by S. T. Hu (13) and by S. Wylie. Following S. Wylie, to whom the concept of a track group is due, we define the group directly by means of maps of P X I into X, which carry (Qxl U Pxl) to x0. Though (P, Q) m depends only on the homotopy type of the pair [P, Q] (that is, not on the particular pair [P, Q] chosen) it is not true in general that it depends only on the homotopy type of [P*5 Q*], where P*, Q* are the m-fold suspensions of P, Q, respectively. If P D Q D R, and P, Q, X satisfy certain conditions, there is an exact sequence